Wick product

In probability theory, the Wick product, named for Italian physicist Gian-Carlo Wick, is a particular way of defining an adjusted product of a set of random variables. In the lowest order product the adjustment corresponds to subtracting off the mean value, to leave a result whose mean is zero. For the higher-order products the adjustment involves subtracting off lower order (ordinary) products of the random variables, in a symmetric way, again leaving a result whose mean is zero. The Wick product is a polynomial function of the random variables, their expected values, and expected values of their products.

The definition of the Wick product immediately leads to the Wick power of a single random variable, and this allows analogues of other functions of random variables to be defined on the basis of replacing the ordinary powers in a power series expansion by the Wick powers. The Wick powers of commonly-seen random variables can be expressed in terms of special functions such as Bernoulli polynomials or Hermite polynomials.

Assume that X₁, ..., X_k are random variables with finite moments. The Wick product

$${\displaystyle \langle X_{1},\dots ,X_{k}\rangle \,}$$

is a sort of product defined recursively as follows:^{[citation needed]}

$${\displaystyle \langle \rangle =1\,}$$

(i.e. the empty product—the product of no random variables at all—is 1). For k ≥ 1, we impose the requirement

$${\displaystyle {\partial \langle X_{1},\dots ,X_{k}\rangle \over \partial X_{i}}=\langle X_{1},\dots ,X_{i-1},{\widehat {X}}_{i},X_{i+1},\dots ,X_{k}\rangle ,}$$

where ${\displaystyle {\widehat {X}}_{i}}$ means that X_i is absent, together with the constraint that the average is zero,

$${\displaystyle \operatorname {E} {\bigl [}\langle X_{1},\dots ,X_{k}\rangle {\bigr ]}=0.\,}$$

Equivalently, the Wick product can be defined by writing the monomial X₁, ..., X_k as a "Wick polynomial":

$${\displaystyle X_{1}\dots X_{k}=\!\!\sum _{S\subseteq \left\{1,\dots ,k\right\}}\!\!\operatorname {E} \left[\textstyle \prod _{i\notin S}X_{i}\right]\cdot \langle X_{i}:i\in S\rangle ,}$$

where ${\displaystyle \langle X_{i}:i\in S\rangle }$ denotes the Wick product ${\displaystyle \langle X_{i_{1}},\dots ,X_{i_{m}}\rangle }$ if ${\displaystyle S=\left\{i_{1},\dots ,i_{m}\right\}.}$ This is easily seen to satisfy the inductive definition.

It follows that

$${\displaystyle {\begin{aligned}\langle X\rangle =&\ X-\operatorname {E} [X],\\[4pt]\langle X,Y\rangle =&\ XY-\operatorname {E} [Y]\cdot X-\operatorname {E} [X]\cdot Y+2(\operatorname {E} [X])(\operatorname {E} [Y])-\operatorname {E} [XY],\\[4pt]\langle X,Y,Z\rangle =&\ XYZ\\&-\operatorname {E} [Y]\cdot XZ\\&-\operatorname {E} [Z]\cdot XY\\&-\operatorname {E} [X]\cdot YZ\\&+2(\operatorname {E} [Y])(\operatorname {E} [Z])\cdot X\\&+2(\operatorname {E} [X])(\operatorname {E} [Z])\cdot Y\\&+2(\operatorname {E} [X])(\operatorname {E} [Y])\cdot Z\\&-\operatorname {E} [XZ]\cdot Y\\&-\operatorname {E} [XY]\cdot Z\\&-\operatorname {E} [YZ]\cdot X\\&-\operatorname {E} [XYZ]\\&+2\operatorname {E} [XY]\operatorname {E} [Z]\\&+2\operatorname {E} [XZ]\operatorname {E} [Y]\\&+2\operatorname {E} [YZ]\operatorname {E} [X]\\&-6(\operatorname {E} [X])(\operatorname {E} [Y])(\operatorname {E} [Z]).\end{aligned}}}$$

In the notation conventional among physicists, the Wick product is often denoted thus:

$${\displaystyle :X_{1},\dots ,X_{k}:\,}$$

and the angle-bracket notation

$${\displaystyle \langle X\rangle \,}$$

is used to denote the expected value of the random variable X.

The nth Wick power of a random variable X is the Wick product

$${\displaystyle X'^{n}=\langle X,\dots ,X\rangle \,}$$

with n factors.

The sequence of polynomials P_n such that

$${\displaystyle P_{n}(X)=\langle X,\dots ,X\rangle =X'^{n}\,}$$

form an Appell sequence, i.e. they satisfy the identity

$${\displaystyle P_{n}'(x)=nP_{n-1}(x),\,}$$

for n = 0, 1, 2, ... and P₀(x) is a nonzero constant.

For example, it can be shown that if X is uniformly distributed on the interval [0, 1], then

$${\displaystyle X'^{n}=B_{n}(X)\,}$$

where B_n is the nth-degree Bernoulli polynomial. Similarly, if X is normally distributed with variance 1, then

$${\displaystyle X'^{n}=H_{n}(X)\,}$$

where H_n is the nth Hermite polynomial.

$${\displaystyle (aX+bY)^{'n}=\sum _{i=0}^{n}{n \choose i}a^{i}b^{n-i}X^{'i}Y^{'{n-i}}}$$

$${\displaystyle \langle \operatorname {exp} (aX)\rangle \ {\stackrel {\mathrm {def} }{=}}\ \sum _{i=0}^{\infty }{\frac {a^{i}}{i!}}X^{'i}}$$

This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (May 2012) (Learn how and when to remove this message)

• Wick Product Springer Encyclopedia of Mathematics
• Florin Avram and Murad Taqqu, (1987) "Noncentral Limit Theorems and Appell Polynomials", Annals of Probability, volume 15, number 2, pages 767—775, 1987.
• Hida, T. and Ikeda, N. (1967) "Analysis on Hilbert space with reproducing kernel arising from multiple Wiener integral". Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66). Vol. II: Contributions to Probability Theory, Part 1 pp. 117–143 Univ. California Press
• Wick, G. C. (1950) "The evaluation of the collision matrix". Physical Rev. 80 (2), 268–272.